Related papers: Carleson-type removability for $p$-parabolic equat…

Let $w\in L^1\_{loc}(\R^n)$ be apositive weight. Assuming that a doubling condition and an $L^1$ Poincar\'e inequality on balls for the measure $w(x)dx$, as well as a growth condition on $w$, we prove that the compact subsets of $\R^n$…

We study local regularity for nonlocal doubly degenerate parabolic equations. The model equation is \begin{equation*}\begin{split}…

We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic $p$-Laplacian. Assuming that such solutions continuously vanish on some distinguished part of…

In this paper, we establish a necessary and sufficient stability condition for a class of two coupled first-order linear hyperbolic partial differential equations. Through a backstepping transform, the problem is reformulated as a stability…

We prove that $L^2$ weak solutions to hypoelliptic equations with bounded measurable coefficients are H\"older continuous. The proof relies on classical techniques developed by De Giorgi and Moser together with the averaging lemma and…

We want to prove a Harnack type inequality for solutions of strongly degenerate parabolic, or elliptic-parabolic, equations. To do that, we first define a De Giorgi class of order $p = 2$ that contains the solutions of evolution equations…

We establish the local H\"older continuity for the nonnegative weak solutions of certain doubly nonlinear parabolic equations possessing a singularity in the time derivative part and a degeneracy in the principal part. The proof involves…

We give necessary conditions for a set E to be removable for Holder continuous quasiregular mappings in the plane. We also obtain some removability results for Holder continuous mappings of finite distortion.

We give a unified proof of H\"{o}lder regularity of weak solutions for mixed local and nonlocal $p$-Laplace type parabolic equations with the full range of exponents $1<p<\infty$. Our proof is based on the expansion of positivity together…

We give a proof of H\"older continuity for bounded local weak solutions to the equation $u_t= \sum_{i=1}^N (|u_{x_i}|^{p_i-2} u_{x_i})_{x_i}$, in $\Omega \times [0,T]$, with $\Omega \subset \subset \mathbb{R}^N$, under the condition $…

Elliptic and parabolic integro-differential model problems are considered in the whole space. By verifying H\"ormander condition, the existence and uniqueness is proved in L_{p}-spaces of functions whose regularity is defined by a scalable,…

We make progress on an interesting problem on the boundedness of maximal modulations of the Hilbert transform along the parabola. Namely, if we consider the multiplier arising from it and restrict it to lines, we prove uniform $L^p$ bounds…

This paper constructs a solvability theory for a system of stochastic partial differential equations. On account of the Kolmogorov continuity theorem, solutions are looked for in certain H\"older-type classes in which a random field is…

We consider the non-degenerate second-order parabolic partial differential equations of non-divergence form with bounded measurable coefficients (not necessary continuous). Under some assumptions it is known that the fundamental solution to…

We prove the H\"{o}lder continuity of sign-changing solutions to the equation of the type $$\frac{\partial}{\partial t}\big(|u|^{q-1} u\big)- div\Big(|D u|^{p-2}\,D u\Big)=0,$$ where numbers $p$, $q$ satisfy the conditions $$0<q<p-1\quad…

The aim of this work is to present the regularity condition (also known in the literature as structure condition) an integro-differential operator may satisfy in order for the domination principle to hold for (sub-,super-) solutions of…

In this paper, we fully resolve the question of whether the Regularity problem for the parabolic PDE $-\partial_tu + \mbox{div}(A\nabla u)=0$ on a Lipschitz cylinder $\mathcal O\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under…

We study stable conditional measures for a certain equilibrium measure for hyperbolic endomorphisms, on basic sets with overlaps; we show that these conditional measures are geometric probabilities and measures of maximal stable dimension.…

We offer a measure-theoretic extension of the concept and theory of $k$-contraction, including their generalization on fractional dimensions $d$. The respective contraction property is defined through the exponential decay of the…

We prove the $W^{1,2}_{p}$-solvability of second order parabolic equations in nondivergence form in the whole space for $p\in (1,\infty)$. The leading coefficients are assumed to be measurable in one spatial direction and have vanishing…